Why does the flame shut off before a hot to warm temperature can be achieved

It sounds like the thermostat that operated gas valve is failing or the gas valve is bad. This is a repair left to professionals. Now I would price a new heater and install it myself. Repairing older heater is not a good idea, the rest of the heater is old and will fail.

Don't know your specific unit, but in general, heating systems are two-stage devices:

1: Its too cold: the motorised HW/CH valve moved to the required position, and on achieving it, the circulaion pump starts and the boiler is enabled. (The circ pump may start immediately - its not usually important.) The hot water from the boiler circulates through the heat exchanger and/or radiators until the thermostat thinks its target is warm enough, whereupon it disables the boiler, stops the pump and closes/relaxes the valve(s).

2: The boiler is off while disabled. When enabled, it heats the circulating water to a 'high' temperature and seeks to keep it there. If it reaches that temperature, the boiler will cycle on and off as necessary to maintain the set 'high' temperature. (The circulation pump must run all the time the boiler is enabled.) This may be what you're noticing. Additionally, some boilers will run the circulation pump for a little while after being disabled to reduce the temperature if the water in the pipes and boiler.

Your question doesn't say how the pump running corresponds to thee thermostat settings, but if it were stuck on, I think you'd know!

Sounds like the water temperature sensor has gone wacky. Since I don't know exactly which unit you have, it could be the sensor, or the board.

has this heater always acted like this? if so
you need to supply heater with the right water pressure or flow
try cutting back by turning inlet valve off to restrict flow or restrict flow on hot end. hope it works it did on mine

1) Bad gas control thermostat, or thermostat has come out of calibration, or water leak running hot water out of tank.
Turn up temperature.
Replace gas control valve.
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html#intellivent
http://waterheatertimer.org/How-to-adjust-temperature-on-water-heater.html

2) Clean air intake, put ear against pipes and listen for water leak, check that flame is clear blue and that combustion gasses are flowing up the vent pipe and not blocked.

3) Call service technician.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Reliance
2nd and 3rd digit of serial number is year
C05 is 2005

What color is flame? Should be clear blue.
Pilot is burning so thermocouple is good, and sitting in pilot flame.
Pilot flame should be clear blue.
Otherwise clean burner parts.

Is water at tank hot??
If burner is not turning on, the gas control thermostat is probably bad.
http://waterheatertimer.org/pdf/How-to-replace-water-heater-gas-valve.pdf

Otherwise more information is needed.
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Get more fresh air to the unit.
Clean the air intake/ flame trap/ flame arrestor located on bottom of tank.
http://waterheatertimer.org/images/Flame-trap-inspection-325.jpg

Post question on pro water heater forum.
http://www.thetankatwaterheaterrescue.com/forums/forum3/

Add a comment with type of gas control valve, for possible troubleshoot.
Copy following link to read service manual by type of gas control valve:
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html#intellivent

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

does the flame stay on all the time? sounds like bad gas valve/thermostat

Have you tried it with both (hot rod and Gas ) turned on at same time? Is tank well insulated? (looking from behind tank.)
Does temperature drop off quickly, or gradually? Did it supply hot water better before, or this is first time you really measured it out? Need to keep in mind that as hot goes out the top fitting to hot lines, cold is coming in the bottom fitting and beginning to mix with hot, which obvioulsy will cause hot to gradually become warm as it enters. 2 gallons out hot tap, 2 gals., at bottom of tank, then mixing as it flows in usually equals, about 2 gal or less of warm, then cold again. Get many complaints of only enough for a short shower when they have 6 gal tank. You can try adjusting temperature up a little to see if it helps. Keep an eye on pressure relief valve dripping excessively as well as shut pump off for 20min-30 min, then turn it back on before touching any taps, to see if it cycles at all, indicating a leak in the system. If pump doesn't cycle when turned back on, no leaks.